Question

prove that (1-cos 2x+sin x)/(sin 2x +cos x)=tan x​

Answer 1

Answer:

we know that

cos (2x) = cos^2(x) - sin^2(x)

= 2,cos^2 (x) -1 = 1- 2sin^2 (x)

and

sin(2x) = 2 sin(x) cos(x)

LHS =

left hand side

$left \: hand \: side \: = \: \frac{(1 - \cos(2x) + \sin(x) )}{ \sin(2x) + \cos(x) } \\ = \frac{(1 - 1 + \: 2 \ { \sin(x) }^{2} + \sin(x)) }{2 \sin(x) \cos(x) + \cos(x) } \\ = \frac{ \sin(x) (2 \sin(x) + 1)}{ \cos(x) (2 \sin(x) + 1) } \\ = \frac{ \sin(x) }{ \cos(x) } = \tan(x) \\ = right \: hand \: side$